Distributed Bearing-Pose Estimation

• Distributed bearing-based pose estimation is a networked observer that uses local bearing measurements and neighbor communication to reconstruct agents’ positions and orientations.
• It employs projection operators and consensus-based update laws that guarantee exponential convergence under infinitesimal bearing rigidity conditions.
• Extensions to the approach include handling directed graphs, time-varying orientations, and hybrid sensing, with anchors or constraints used to resolve inherent gauge freedoms.

to=arxiv_search.search 大发时时彩计划 code : {"query":"distributed bearing-based pose estimation multi-agent network localization bearing rigidity distributed observer arXiv", "max_results": 10} to=arxiv_search.search 天天中彩票网站 code : {"query":"(Zhao et al., 2015) OR (Boughellaba et al., 2023) OR (Le-Phan et al., 2023) OR (Presenza et al., 2 Jun 2026) OR (Zelazo et al., 2013) OR (Wang et al., 2024) OR (Huiming et al., 2023)", "max_results": 10} to=arxiv_search.search ചികിത code : {"query":"Bearing-Based Distributed Control and Estimation of Multi-Agent Systems", "max_results": 5} A distributed bearing-based pose estimator is a networked observer that reconstructs the positions, or the positions and orientations, of agents from local bearing measurements and neighbor-to-neighbor communication. In the canonical anchor-based formulation, each follower maintains a position estimate, a subset of anchors knows its true absolute position, and the update law uses orthogonal projections of inter-agent estimate differences onto subspaces orthogonal to measured bearings; under infinitesimal bearing rigidity and sufficiently many anchors, the estimator converges exponentially to the true configuration and removes the translation-and-scale ambiguity inherent to bearing-only sensing (Zhao et al., 2015). Subsequent work extends the same basic idea to directed acyclic graphs with time-varying orientations, randomized gossip updates, $SE(2)$ body-frame formulations, angle-rigid directed sensing graphs, certifiable convex relaxations, and hybrid bearing–distance–inertial localization architectures (Boughellaba et al., 2023).

1. State, measurement, and graph models

In the anchor-based network-localization model, each agent $i$ has a true constant position $p_i\in\mathbb R^d$ and maintains an estimate $\hat p_i(t)\in\mathbb R^d$. The estimator is written as a continuous-time single integrator,

$\dot{\hat p}_i(t)=u_i(t),$

on a fixed, undirected, connected graph $\mathcal G=(\mathcal V,\mathcal E)$. If $(i,j)\in\mathcal E$, agents $i$ and $j$ can measure the bearing

$$g_{ij}=\frac{p_j-p_i}{\|p_j-p_i\|}\in\mathbb R^d,\qquad g_{ji}=-g_{ij},$$

and the basic projection operator is

$i$0

which projects onto the subspace orthogonal to $i$1 (Zhao et al., 2015).

The same family of estimators appears in dynamic pose problems. In one $i$2 formulation, the positions $i$3 are fixed but the orientations $i$4 evolve according to

$i$5

where $i$6 is the body-measured angular velocity. Agent $i$7 measures only the local time-varying bearing

$i$8

and exchanges $i$9 with its neighbors over a directed graph (Boughellaba et al., 2023).

A body-frame formulation in $p_i\in\mathbb R^d$0 uses local bearing angles

$p_i\in\mathbb R^d$1

or, equivalently, unit vectors

$p_i\in\mathbb R^d$2

with $p_i\in\mathbb R^d$3 rotating a global vector into the agent’s body frame (Zelazo et al., 2013). A more recent orientation-free formulation does not estimate bearings themselves but angles $p_i\in\mathbb R^d$4 computed from pairs of body-frame bearings, and then recovers orientations afterward from estimated positions, bearings, and bearing derivatives (Presenza et al., 2 Jun 2026).

These models differ in state dimension and graph orientation, but they share a common structure: geometric information enters through direction constraints, while distributed estimation is achieved through local exchanges of estimates and local projection or gradient terms.

2. Rigidity, gauge freedoms, and observability

Bearing-only sensing does not determine an absolute configuration without additional structure. In the static anchor-based problem, infinitesimal bearing rigidity is the key identifiability condition. If the bearing-rigidity matrix is

$p_i\in\mathbb R^d$5

then infinitesimal bearing rigidity is equivalent to

$p_i\in\mathbb R^d$6

The nullspace shows that all bearings are preserved by global translations and uniform scaling. Accordingly, once all inter-neighbor bearings match the true ones, the configuration is unique only up to translation and scale; anchors pin down those remaining degrees of freedom (Zhao et al., 2015).

In $p_i\in\mathbb R^d$7, the corresponding infinitesimal-rigidity criterion is

$p_i\in\mathbb R^d$8

where $p_i\in\mathbb R^d$9 is the directed bearing-rigidity matrix. Its four trivial infinitesimal motions are two global translations in $\hat p_i(t)\in\mathbb R^d$0, a uniform dilation, and a coordinated rotation in which each agent spins in place while the whole configuration rotates in $\hat p_i(t)\in\mathbb R^d$1 so that all local bearings remain constant (Zelazo et al., 2013).

Angle-rigidity theory introduces a different observability notion. For the angle-rigidity matrix $\hat p_i(t)\in\mathbb R^d$2, the nullspace always contains the seven-dimensional space of infinitesimal similarities,

$\hat p_i(t)\in\mathbb R^d$3

A framework is infinitesimally angle-rigid if

$\hat p_i(t)\in\mathbb R^d$4

The method based on angle rigidity states explicitly that this is a weaker condition than bearing rigidity and that fewer edges, and no reciprocity, are required to attain full infinitesimal rank (Presenza et al., 2 Jun 2026).

A common misconception is that bearing measurements alone identify absolute pose. The rigidity results show the opposite: bearings determine a formation only modulo gauge freedoms, and each formulation must remove those freedoms explicitly through anchors, penalties, range pins, leader states, or additional geometric constraints.

3. Canonical anchor-based distributed estimator

The anchor-based estimator for static positions is the prototype of the distributed bearing-based pose estimator. The node set is partitioned into anchors $\hat p_i(t)\in\mathbb R^d$5 and followers $\hat p_i(t)\in\mathbb R^d$6. For anchors, $\hat p_i(t)\in\mathbb R^d$7. For each follower $\hat p_i(t)\in\mathbb R^d$8, the local update law is

$\hat p_i(t)\in\mathbb R^d$9

while $\dot{\hat p}_i(t)=u_i(t),$0 for anchors (Zhao et al., 2015).

Stacking the anchor and follower estimates as $\dot{\hat p}_i(t)=u_i(t),$1, one defines the bearing-Laplacian $\dot{\hat p}_i(t)=u_i(t),$2 and partitions it as

$\dot{\hat p}_i(t)=u_i(t),$3

The estimator then becomes

$\dot{\hat p}_i(t)=u_i(t),$4

This is a linear consensus-type protocol with matrix weights $\dot{\hat p}_i(t)=u_i(t),$5. Each agent needs only its neighbors’ estimates and the fixed true bearings $\dot{\hat p}_i(t)=u_i(t),$6 (Zhao et al., 2015).

Its convergence analysis is entirely spectral. Under an undirected, fixed, connected graph, infinitesimal bearing rigidity, and $\dot{\hat p}_i(t)=u_i(t),$7, the bearing-Laplacian satisfies

$\dot{\hat p}_i(t)=u_i(t),$8

and the principal submatrix $\dot{\hat p}_i(t)=u_i(t),$9 is positive definite if and only if $\mathcal G=(\mathcal V,\mathcal E)$0. The resulting theorem states that, for any initial $\mathcal G=(\mathcal V,\mathcal E)$1,

$\mathcal G=(\mathcal V,\mathcal E)$2

exponentially fast. The proof uses the unique equilibrium of the linear system and the identity $\mathcal G=(\mathcal V,\mathcal E)$3, which follows from $\mathcal G=(\mathcal V,\mathcal E)$4 (Zhao et al., 2015).

The discrete-time forward-Euler implementation is

$\mathcal G=(\mathcal V,\mathcal E)$5

with stability condition

$\mathcal G=(\mathcal V,\mathcal E)$6

Followers may start from any $\mathcal G=(\mathcal V,\mathcal E)$7; the stated implementation remark is that there is no special requirement other than not exactly collocated with anchors in a degenerate configuration. Each update costs $\mathcal G=(\mathcal V,\mathcal E)$8 flops, and the memory cost is $\mathcal G=(\mathcal V,\mathcal E)$9 (Zhao et al., 2015).

The simulation example in $(i,j)\in\mathcal E$0 uses 50 agents, 269 edges, and 4 anchors, with $(i,j)\in\mathcal E$1. The initial follower estimates are chosen uniformly at random, the performance metric is

$(i,j)\in\mathcal E$2

and all $(i,j)\in\mathcal E$3 converge exponentially to zero. The reported interpretation is that the example demonstrates global convergence, robustness to large initial errors, and correctness of the protocol (Zhao et al., 2015).

4. Kinematic, asynchronous, and large-scale variants

For mobile agents with known velocity inputs, the projection structure can be embedded into a kinematic observer. In the simultaneous localization and affine formation tracking setting, agent $(i,j)\in\mathcal E$4 evolves according to $(i,j)\in\mathcal E$5, with heading $(i,j)\in\mathcal E$6. The distributed position-estimate dynamics are

$(i,j)\in\mathcal E$7

where $(i,j)\in\mathcal E$8 for anchors and $(i,j)\in\mathcal E$9 otherwise. For the estimation error $i$0, the stacked dynamics are

$i$1

With the Lyapunov function $i$2, the paper states

$i$3

so $i$4 exponentially fast at rate $i$5 (Huiming et al., 2023).

The same work isolates estimator performance in a 7-agent example with anchors $i$6 under two scenarios: translation along a sinusoid with constant desired bearings, and circular rotation with time-varying desired bearings. In both cases $i$7 falls from an $i$8 initial value to below $i$9 in $j$0–$j$1 with $j$2, and the log-scale plots confirm essentially exponential decay (Huiming et al., 2023).

Asynchronous operation is addressed by randomized gossip localization. At each discrete time, a node wakes up uniformly at random, selects a neighbor with positive probability, and exactly that pair interacts. If both are followers, the pairwise update is

$j$3

$j$4

while if one node is a beacon, only the follower updates. In expectation the follower error satisfies

$j$5

The basic step-size requirement is

$j$6

and a stronger condition gives geometric mean-square decay and almost-sure convergence of every follower estimate to its true position (Le-Phan et al., 2023).

The large-scale simulation for this gossip scheme uses $j$7 nodes in $j$8, two beacons, and $j$9 gossip updates. The total bearing error

$$g_{ij}=\frac{p_j-p_i}{\|p_j-p_i\|}\in\mathbb R^d,\qquad g_{ji}=-g_{ij},$$0

decays exponentially to zero. The communication cost per interaction is $$g_{ij}=\frac{p_j-p_i}{\|p_j-p_i\|}\in\mathbb R^d,\qquad g_{ji}=-g_{ij},$$1 scalars, and the computation cost per agent is $$g_{ij}=\frac{p_j-p_i}{\|p_j-p_i\|}\in\mathbb R^d,\qquad g_{ji}=-g_{ij},$$2 flops (Le-Phan et al., 2023).

These variants preserve the same geometric core—projection onto subspaces orthogonal to measured bearings—while changing the activation model, the kinematics, or the scale of the network.

5. Distributed full-pose estimation

When orientations are unknown and time-varying, distributed bearing-based pose estimation becomes a cascaded nonlinear observer problem. In one $$g_{ij}=\frac{p_j-p_i}{\|p_j-p_i\|}\in\mathbb R^d,\qquad g_{ji}=-g_{ij},$$3 formulation, the graph is directed and acyclic, agents $$g_{ij}=\frac{p_j-p_i}{\|p_j-p_i\|}\in\mathbb R^d,\qquad g_{ji}=-g_{ij},$$4 and $$g_{ij}=\frac{p_j-p_i}{\|p_j-p_i\|}\in\mathbb R^d,\qquad g_{ji}=-g_{ij},$$5 are leaders, and each follower has at least two non-collinear bearings. The follower attitude observer is

$$g_{ij}=\frac{p_j-p_i}{\|p_j-p_i\|}\in\mathbb R^d,\qquad g_{ji}=-g_{ij},$$6

with leaders setting $$g_{ij}=\frac{p_j-p_i}{\|p_j-p_i\|}\in\mathbb R^d,\qquad g_{ji}=-g_{ij},$$7. The attitude error is $$g_{ij}=\frac{p_j-p_i}{\|p_j-p_i\|}\in\mathbb R^d,\qquad g_{ji}=-g_{ij},$$8, and the unforced subsystem is shown to be almost-globally asymptotically stable, while the forced system satisfies an ISS inequality of the form

$$g_{ij}=\frac{p_j-p_i}{\|p_j-p_i\|}\in\mathbb R^d,\qquad g_{ji}=-g_{ij},$$9

Induction along the acyclic ordering yields network-wide AGAS of the attitude observer (Boughellaba et al., 2023).

The position observer in the same paper is

$i$00

with error $i$01. The unforced position subsystem is globally exponentially stable, and the ISS estimate

$i$02

leads to the overall AGAS result for $i$03. The simulation places eight agents at the corners of the cube $i$04, chooses bounded sinusoidal angular velocities, and shows the average attitude and position errors decaying smoothly to zero (Boughellaba et al., 2023).

A different route to full-pose estimation is the distributed $i$05 estimator based on a directed bearing-rigidity matrix and gradient descent on an augmented cost

$i$06

A root agent $i$07 and a reference neighbor $i$08 are chosen so that $i$09 is known, or set to $i$10 w.l.o.g. The update depends only on each agent’s own estimate and the current estimates of its outgoing neighbors, so the scheme is fully distributed. Under $i$11, the added penalty terms remove the trivial motions and standard gradient-descent arguments give local asymptotic convergence (Zelazo et al., 2013).

More recently, a time-varying $i$12-D observer has been proposed that estimates positions from angles computed from body-frame bearings without using orientations in the position subsystem. Its position observer is

$i$13

and orientations are then recovered on $i$14 by gradient flows derived from local fitting costs. The stated graph condition is infinitesimal angle rigidity rather than bearing rigidity, and the overall result is local uniform exponential stability under persistently exciting motions for a subset of robots (Presenza et al., 2 Jun 2026).

Taken together, these formulations show that orientation need not be handled in a single way: it may be estimated jointly with position in a cascade, fixed by a rooted $i$15 gauge choice, or recovered after an orientation-free position step.

6. Closed-form, certifiable, and hybrid directions

A related development is the closed-form $i$16-DoF inter-robot pose estimator that assumes roll and pitch are known from VIO and estimates only yaw and translation. Its state is

$i$17

Yaw is obtained from a relaxed homogeneous linear system $i$18, followed by projection of each $i$19 onto the unit circle via

$i$20

and translation is recovered by tangent-plane projection

$i$21

and a Total Least Squares solve. The observability analysis states that $i$22, so global translation and global yaw shift are always unobservable, and identifies two additional degeneracies: collinear formations and shape-preserving formations. The autonomous observability test monitors $i$23 and $i$24 and triggers the solution only when geometry has stabilized and excitation is sufficient (De et al., 25 Jun 2026).

The same work reports simulations with $i$25–$i$26 robots and real-world experiments with $i$27 quadrotors. In the real experiments, Ours-OT achieves approximately $i$28 rotation error, approximately $i$29 translation error, runtime approximately $i$30, and estimation latency as early as $i$31–$i$32, while the observability test eliminates reliance on a predefined fixed-length sliding window (De et al., 25 Jun 2026).

Certifiable mutual localization formulates bearing-only pose recovery as a maximum-likelihood problem over $i$33, relaxes it to an SDP,

$i$34

and then drops the rank constraint to obtain a convex program. The paper also gives a distributed ADMM sketch and a certificate matrix $i$35 such that global optimality is certified by

$i$36

with $i$37 reflecting the world-frame gauge. Under bounded noise, the condition

$i$38

strictly guarantees estimation optimality (Wang et al., 2024).

A broader hybrid direction is CREPES-X, which combines bearing, distance, and inertial sensing in a two-stage hierarchy. Its single-frame closed-form estimator uses MDS, Wahba alignment, yaw estimation from horizontal projections, and a graph-based bearing outlier rejector; its multi-frame stage uses IMU pre-integration, robocentric relative kinematics, a loosely coupled initialization, and a tightly coupled sliding-window optimization. On a 10-robot benchmark with window 10, the reported accuracies are $i$39 for SFC, $i$40 for SFO, $i$41 for MFLO, and $i$42 for MFTO; the real-world indoor result for 5 robots is $i$43 and $i$44 RMSE, with stability up to $i$45 bearing outliers and real-time operation up to approximately 20 robots on desktop (Li et al., 31 Dec 2025).

Across these lines of work, the phrase “distributed bearing-based pose estimator” no longer denotes a single algorithmic template. It now includes projection-based linear localization, nonlinear cascaded observers on $i$46, gradient systems on $i$47, angle-rigidity observers on directed graphs, pairwise gossip schemes, closed-form partial-pose solvers with observability tests, and certifiable convex programs. What remains invariant is the geometric premise: relative direction information is sufficient for pose recovery only when the sensing graph and the agent motions satisfy the appropriate rigidity or observability conditions, and distributed estimation succeeds only when those geometric conditions are matched by a gauge-fixing mechanism and a stability proof suited to the chosen state space.